## The Annals of Statistics

- Ann. Statist.
- Volume 15, Number 1 (1987), 278-295.

### Asymptotic Expansions in Anscombe's Theorem for Repeated Significance Tests and Estimation after Sequential Testing

#### Abstract

Let $x_1, x_2, \cdots$ be independent and normally distributed with unknown mean $\theta$ and variance 1. Let $\tau = \inf \{n \geq 1: |s_n| \geq \sqrt{2a(n + c)}\}$. Then a repeated significance test for a normal mean rejects the hypothesis $\theta = 0$ if and only if $\tau \leq N_0$ for some positive integer $N_0$. The problem we consider is estimation of $\theta$ based on the data $x_1,\cdots, x_T, T = \min\{\tau, N_0\}$. We shall solve this problem by obtaining the asymptotic expansion of the distribution of $(s_\tau - \tau\theta)/\sqrt{\tau}$ as $a \rightarrow \infty$, and then constructing the confidence intervals for $\theta$.

#### Article information

**Source**

Ann. Statist., Volume 15, Number 1 (1987), 278-295.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176350266

**Digital Object Identifier**

doi:10.1214/aos/1176350266

**Mathematical Reviews number (MathSciNet)**

MR885737

**Zentralblatt MATH identifier**

0615.62107

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 62L12: Sequential estimation 60K05: Renewal theory

**Keywords**

Sequential estimation sequential test confidence interval Anscombe's theorem

#### Citation

Takahashi, Hajime. Asymptotic Expansions in Anscombe's Theorem for Repeated Significance Tests and Estimation after Sequential Testing. Ann. Statist. 15 (1987), no. 1, 278--295. doi:10.1214/aos/1176350266. https://projecteuclid.org/euclid.aos/1176350266